[Weinber QFT] 15.2 Gauge Theory Lagrangians and Simple Lie Groups

  This article is one of the posts in the Textbook Commentary Project.

 The transformation rules of the gauge-field tensor $F^\alpha_{\mu\nu}$ and the matter fields $\phi$ and their gauge-covariant derivatives do not involve the derivatives of the transformation parameters $\epsilon^\alpha(x)$, so if the Lagrangian is constructed solely from these ingredients, and if it is invariant under global transformations with $\epsilon^\alpha$ constant, then it is invaritant under gauge transformations with general position-dependent $\epsilon^\alpha(x)$. We therefore assume that the Lagrangian satisfies these conditions: that is, $$\begin{align} \mathcal{L} = \mathcal{L} (\psi, D_\mu \psi, D_\nu D_\mu \psi, \cdots, F^\alpha_{\mu\nu}, D_\rho F^\alpha_{\mu\nu} \cdots)\end{align}$$ with the invariance condition: $$\begin{align} \frac{\partial \mathcal{L}}{\partial \psi_l} i(t_\alpha)_l^m\psi_m + \frac{\partial \mathcal{L}}{\partial (D_\mu \psi_l)} i(t_\alpha)_l^m (D_\mu \psi_m) + \frac{\partial \mathcal{L}}{\partial (D_\nu D_\mu \psi_l)} i(t_\alpha)_l^m (D_\nu D_\mu \psi_m) + \cdots + \frac{\partial \mathcal{L}}{\partial F^\beta_{\mu\nu}} C^\beta_{\gamma \alpha} F^\gamma_{\nu\mu} + \frac{\partial \mathcal{L}}{\partial D_\rho F^\beta_{\mu\nu}} C^\beta_{\gamma\alpha} D_\rho F^\gamma_{\nu\mu}+\cdots = 0.\end{align}$$

On the other hand, the Lagrangian may not depend on the gauge field itself, except insofar as it appears in $F^\alpha_{\mu\nu}$ and in gauge-covariant derivatives $D_\mu$. In particular, a mass term- $\frac{1}{2} m^2_{\alpha\beta} A_{\alpha\mu}A_\beta^\mu$ is ruled out.

We shall concentrate now on the terms in the Lagrangian that depend only on $F^\alpha_{\mu\nu}$. Just as in electrodynamics, for any massless particle of unit spin the Lagrangian must contain a free-particle term quadratic in $\partial_\mu A^\alpha_\nu - \partial_\nu A^\alpha_\mu$, and gauge invariance then dictates that this free-particle term should appear as part of a term quadratic in the field-strength tensor $F^\alpha_{\mu\nu}$. Lorentz invariance and parity conservation dictate its form as $$\begin{align} \mathcal{L}_A = -\frac{1}{2} g_{\alpha\beta} F^\alpha_{\mu\nu} F^{\beta\mu\nu}\end{align}$$ with a constant matrix $g_{\alpha\beta}$. If we do not assume parity (or CP or T) conservation, then we may also include in the Lagrangian a term $$\begin{align} \mathcal{L}'_A = -\frac{1}{2} \theta_{\alpha\beta} e^{\mu\nu\rho\sigma}F^\alpha_{\mu\nu} F^\beta_{\rho\sigma}\end{align}$$ with another constant matrix $\theta_{\alpha\beta}$. This term is actually a dericative, and therefore does not affect the field equations or the Feynman rules. Such a term would, however, have non-perturbative quantum mechanical effects, to be discussed in Section 23.6.

Before going on to consider the properties of the matirx $g_{\alpha\beta}$, it is worth drawing attention to the fact the it is not possible to introduce a kinematic term for the gauge field $A^\alpha_\mu(x)$ without also including interactions, the term in Eq. (3) arising from the quadratic part of the field strength $F^\alpha_{\mu\nu}$ defined by Eq. (15.1.13). This is one more respect in which non-Abelian gauge theories resemble general relativity, where the kinematic part of the Lagrangian for the gravitational field is contained in the Einstein-Hilbert Lagrangian density- $\sqrt{g}R/8\pi G$, which also contains self-interactions of the field. The reasons in both cases are similar: the gravitational field interacts with itself because it interacts with anything that carries energy and momentum, and the gauge field interacts with itself because it interacts with anything that transforms according to a non-trivial representation (in this case the adjoint representation) of the gauge group. This is in contrast to the case of electrodynamics, where the photon does not carry electric charge, the quantum number with which it interacts, and it is consequently possible to introduce a kinematic term $-\frac{1}{4} F_{\mu\nu}F^{\mu\nu}$ for the electromagnetic field that does not entail interactions.

The numerical matrix $g_{\alpha\beta}$ may be taken symmetric, and must be taken real to give a real Lagrangian. In order for this term to satisfy the gauge-invariance requirement (2), we muse have for all $\delta$: $$g_{\alpha\beta} F^\alpha_{\mu\nu}C^\beta_{\gamma\delta}R^{\gamma\mu\nu}=0.$$

In order for this to be true without having to impose any functional relations among the $F$s, the matrix $g_{\alpha\beta}$ muse satisfy the condition: $$\begin{align} g_{\alpha\beta}C^\beta_{\gamma\delta}=-g_{\gamma\beta}C^\beta_{\alpha\delta}.\end{align}$$

There is one more important condition on the matrix $g_{\gamma\beta}$. Just as in quantum electrodynamics, the rules of canonical quantization and the positivity properties of the quantum mechanical scalar product require that the matrix $g_{\alpha\beta}$ in the Lagrangian (3) must be positive-definite. (That is, $g_{\alpha\beta}u^\alpha u^\beta$ is positive for all real $u$, and vanishes for some real $u$ only if $u^\alpha=0$ for all $\alpha$.) This is analogous to the requirement that in the kinematic Lagrangian- $\frac{1}{2}Z\partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2$ for a real scalar field $\phi$, the constant $Z$ must be positive-definite.

These requirements on the matrix $g_{\alpha\beta}$ have far reaching implications. They form one of a set of three equivalent conditions: 

a: There exists a real symmetric positive-definte matrix $g_{\alpha\beta}$ that satisfies the invariance condition (4).

b: There is a basis for the Lie algebra (that is, a set of generators $\tilde{t}_\alpha=\mathcal{J}_{\alpha\beta}t_\beta$, with $\mathcal{J}$ a real non-singular matrix) for which the structure constants $\tilde{C}^\alpha_{\beta\gamma}$ are antisymmetric not only in the lower indices $\beta$ and $\gamma$ but in all three indices, $\alpha,\beta$ and $\gamma$. (In this basis it is convenient to drop the distinction between upper and lower indices $\alpha,\beta$, etc., and write $\tilde{C}_{\alpha\beta\gamma}$ in place of $\tilde{C}^\alpha_{\beta\gamma}$.)

c: The Lie algebra is the direct sum of commuting compact simple and $U(1)$ subalgebras.

Appendix A of this chapter presents a proof of the equivalence of the conditions a, b, and c.

Before going on to discuss the physical implications of this result, it will be useful to say a bit more about the condition of compactness. We will not use this here, but a compact Lie algebra consists of the generators of a compact Lie group: one for which the invariant volume of the group is finite. For instance, the rotation group is compact; the Lorentz group is not. As a simple example of a simple Lie algebra that is not compact, consider the commutation relations $$[t_1,t_2]=-it_3,\ [t_2,t_3]=it_1,\ [t_3,t_1]=it_2.$$

The structure constant there is real, but not completely antisymmetric; its non-vanishing components are $$C^3_{12}=-C^3_{21}=-1,\ C^1_{23}=-C^1_{32}=1,\ C^2_{31}=-C^2_{13}=1.$$

The metric given by Eq. (15.A.10) is here diagonal, with elements: $$g_{11}=g_{22}=-g_{33}=-2.$$

This is not a positive matrix, so the Lie algebra is not compact. It is in fact the Lie algebra of a non-compact group $O(2,1)$, the Lorentz group in two space and one time dimensions.

Two sets of generators that differ by a real non-singular linear transformation are considered to span the same Lie algebra, and generate the same group. This is not true for complex linear transformations of generators. In particular, any simple Lie algebra can be put into a compact form by a change of phase of the generators in a suitable basis. For instance, for the Lie algebra of the above example, it is only necessary to define new generators $t'_1=it_1$, $t'_2=it_2$, $t'_3=t_3$, for which the commutation relations are $$[t'_1,t'_2]=it'_3,\ [t'_2,t'_3]=it'_1,\ [t'_3,t'_1]=it'_2.$$

The structure constant is now real and totally antisymmetric: $C^a_{bc}=\epsilon_{abc}$. Here $g_{ab}=2\delta_{ab}$, and the algebra is compact. We recognize this, of course, as the familiar algebra of the compact group $O(3)$ of rotations in three dimensions. To see that his is always possible for any simple Lie algebra, note that the matrix $g_{ab}$ defined by Eq. (15.A.10) is real, symmetric, and non-singular, so that by a real orthogonal transformation it may be put in a diagonal form with non-zero elements along the main diagonal. It is then only necessary to multiply all the generators that correspond in this basis to the negative diagonal elements of $g_{ab}$ by factors $i$.

We note without proof that the finite-dimensional representations of compact Lie groups are all unitary, and the finite-dimensional representations of compact Lie algebras are correspondingly all Hermitian. Furthermore, it is easy to see that the only Lie algebras that can have any non-trivial representation by independent finite-dimensional Hermitian matrices $t_\alpha$ are direct sums of $U(1)$ and compact simple Lie algebras. To show this, we may simply define $$g_{\alpha\beta}\equiv \mbox{Tr}\{ t_\alpha t_\beta\} .$$

This matrix is obviously positive-definite, because $g_{\alpha\beta}u^\alpha u^\beta = \mbox{Tr} \{ (u^\alpha t_\alpha )^2 \}$ is positive for any real $u^\alpha$ and vanishes only if $u^\alpha t_\alpha=0$, which is not possible unless all $u^\alpha$ vanish because the $t^\alpha$ are assumed independent. Furthermore this $g_{\alpha\beta}$ satisfies the invariance condition (4), as can be seen by multiplying the commutation relation (2) with $t_\delta$ and taking the trace; this gives $$i C^\gamma_{\alpha\beta} \mbox{Tr}\{ t_\gamma t_\delta \} = \mbox{Tr} \{ [ t_\alpha, t_\beta ] t_\delta \} = \mbox{Tr} \{ t_\delta t_\alpha t_\beta - t_\beta t_\alpha t_\gamma \} ,$$ which is oviously antisymmetric in $\beta$ and $\delta$. Having verified a, we can rely on the above theorem to infer condition c, so that the Lie algebra must be a direct sum of compact simple and $U(1)$ subalgebras.

Let's now return to the physics of gauge theories. In this section we have inferred the existence of a positive symmetric real matrix $g_{\alpha\beta}$, that satisfies the invariance condition (4), from the necessity of constructing a suitable kinematic term in the Lagrangian for the gauge field, and in Appendix A of this chapter we have shown that this result is equivalent to a condition on the Lie algebra, that it is a direct sum of compact simple and $U(1)$ subalgebras. For our purposes, the important thing about this result is that the simple Lie algebras are all of certain limited types and dimensionalities. For instance, it is easy to see that these is no simple Lie algebra with less than three generators, because in one or two dimensions there can be non-zero totally antisymmetric structure constants with three indices. With three generators, an invariant subalgebra can be avoided by taking  $C^3_{12}, C^2_{31}$, and $C^1_{23}$ all non-zero. In the basis in which the structure constant is real and totally antisymmetric, there is obviously only one possibility: $$C_{\alpha\beta\gamma}=c\epsilon_{\alpha\beta\gamma}.$$

Here $c$ is an arbitrary non-zero real constant, which can be eliminated by a change of scale of the generators, $t_\alpha \rightarrow t_\alpha /c$, so the Lie algebra is $$[t_\alpha, t_\beta]=i\epsilon_{\alpha \beta \gamma} t_\gamma .$$

This may be recognized as the Lie algebra of the three-dimensional rotation group $O(3)$, and also of the group $SU(2)$ of unitary unimodular matrices in two dimensions, and was used as a basis for the original non-Ableian gauge theory of Yang and Mills. Continuing in the same way, it can be shown that there are no simple Lie algebras with 4,5,6, or 7 generators, one with 8 generators, and so on. Mathematicians (notably Killing and E. Cartan) have been able to catalog all simple Lie algebras. The compact forms of the simple Lie algebras form several infinite classes of algebras of the 'classical' Lie groups- the unitary unimodular, unitary orthogonal, and unitary symplectic groups -plus just five exceptional Lie algebras. This catalog is presented in Appendix B of this chapter.

It is also shown in Appendix A that under the equivalent conditions a, b, or c, the metric takes the form $$\begin{align} g_{ma,nb}=g_m^{-2}\delta_{mn}\delta_{ab} \end{align}$$ with real $g_m$, where $m$ and $n$ label the simple or $U(1)$ subalgebras, and $a$ and $b$ label the individual generators of these subalgebras. We can eliminate the constants $g_m^{-2}$ by a rescaling of the gauge fields $$\begin{align} A^\mu_{ma} \rightarrow \tilde{A}^\mu_{ma}\equiv g_m^{-1} A^\mu_{ma},\end{align}$$ but then in order to keep the same formulas (15.1.10) and (15.1.13) for $D_\mu \phi$ and $F_\alpha^{\mu\nu}$, we must also redefine the matrices $t_\alpha$ and the structure constants $$\begin{align} t_{ma} \rightarrow \tilde{t}_{ma} = g_m t_{ma},\\ C^{(m)}_{cab} \rightarrow \tilde{C}^{(m)}_{cab} = g_m C^{(m)}_{cab}. \end{align}$$

That is, we can always define the scale of the gauge fields (now dropping the tildes) so that $g_m$ in Eq. (5) is unity: $$\begin{align} g_{\alpha\beta}=\delta_{\alpha\beta}, \end{align}$$ but then the transformation matrices $t_\alpha$ and the structure constants $C_{\alpha\beta\gamma}$ contain an unknown multiplicative factor $g_m$ for each simple or $U(1)$ subalgebra. These factors are the coupling constants of the gauge theory. Alternatively, it is sometimes more convenient to adopt some fixed though arbitrary normalization for the $t_\alpha$ and structure constants within each simple or $U(1)$ subalgebra, in which case the coupling constants appear in the gauge-filed Lagrangian (3) as the factors $g_m^{-2}$ in Eq, (5).